Bernoulli numbers introduced by Jacques Bernoulli (1654-1705), in the second
part of his treatise published in 1713, Ars conjectandi, at the time,
Bernoulli numbers were used for writing the infinite series expansions of
hyperbolic and trigonometric functions. Van den berg was the first to
discuss finding recurrence formulae for the Bernoulli numbers with arbitrary
sized gaps (1881). Ramanujan showed how gaps of size 7 could be found, and
explicitly wrote out the recursion for gaps, of size 6. Lehmer in 1934
extended these methods to Euler numbers, Genocchi numbers, and Lucas numbers
(1934), and calculated the 196-th Bernoulli numbers. The study of
generalized Bernoulli, Euler and Genocchi numbers and polynomials and their
combinatorial relations has received much attention [1]}, [2]}, [3]}-[4]}, [5]}, [6]}, [7]}, [8]}, [9]}. Generalized Bernoulli
polynomials, generalized Euler polynomials and generalized Genocchi numbers
and polynomials are the signs of very strong bond between elementary number
theory, complex analytic number theory, Homotopy theory (stable Homotopy
groups of spheres), differential topology (differential structures on
spheres), theory of modular forms (Eisenstein series), \(p\) -adic analytic
number theory (\(p\) -adic \(L\) -functions), quantum physics(quantum Groups). \(p\) -adic numbers were invented by Kurt Hensel around the end of the nineteenth
century. In spite of their being already one hundred years old, these
numbers are still today enveloped in an aura of mystery within the
scientific community. The \(p\) -adic integral was used in mathematical
physics, for instance, the functional equation of the \(q\) -zeta function, \(q\) -stirling numbers and \(q\) -Mahler theory of integration with respect to the
ring \(\mathbb {Z}_{p}\)  together with Iwasawa's \(p\) -adic \(q\) -\(L\)  functions. Also the \(p\) -adic
interpolation functions of the Bernoulli and Euler polynomials have been
treated by Tsumura [10]} and Young [11]}. Professor T.Kim
[12]}-[13]} also studied on \(p\) -adic interpolation functions
of these numbers and polynomials. In [14]}, Carlitz originally
constructed \(q\) -Bernoulli numbers and polynomials. These numbers and
polynomials are studied by many authors (see cf. [12]}-[16]},
[17]}, [18]}, [19]}). In the last decade, a
surprising number of papers appeared proposing new generalizations of the
Bernoulli, Euler and Genocchi polynomials to real and complex variables. In
[12]}-[21]}, Kim studied some families of multiple Bernoulli,
Euler and Genocchi numbers and polynomials. By using the fermionic \(p\) -adic
invariant integral on \(\mathbb {Z}_{p}\) , he constructed \(p\) -adic Bernoulli, Euler and Genocchi numbers and
polynomials of higher order. A unification (and generalization) of Bernoulli
polynomials and Euler polynomials with a,b and c parameters first was
introduced and investigated by Q.-M.Luo [3]}, [23]}, [24]}, [4]}. After he with H.M.Srivastava defined unification (and
generalization) of Apostol type Bernoulli polynomials with a, b and c
parameters of higher order [4]}. After Hacer Ozden et al . unified and extended the generating functions of the generalized
Bernoulli polynomials, the generalized Euler polynomials and the generalized
Genocchi polynomials associated with the positive real parameters a and b
and the complex parameter. Also they by applying the Mellin transformation
to the generating function of the unification of Bernoulli, Euler and
Genocchi polynomials, constructed a unification of the zeta functions.
Actually their definition provides a generalization and unification of the
Bernoulli, Euler and Genocchi polynomials and also of the
Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials, which
were considered in many earlier investigations by (among others) Srivastava
et al. [27]}, [28]}, [29]},
Karande [30]}. Also they by using a Dirichlet character defined
unification of the generating functions of the generalized Bernoulli, Euler
and Genocchi polynomials and numbers . T. Kim in [31]}, constructed
Apostol-Euler numbers and polynomials by using fermionic expression of \(p\) -adic \(q\) -integral at \(q=-1\) . In this paper by his method we derive several
properties for unification of the multiple twisted Euler and Genocchi
numbers and polynomials.
